Entropy and Information jump for log-concave vectors

TL;DR

This paper extends the understanding of entropy and Fisher information jumps under convolution for log-concave vectors, broadening previous results to non-identically distributed vectors and establishing a quantitative Blachmann-Stam inequality.

Contribution

It generalizes existing entropy jump results to arbitrary log-concave vectors and introduces a related Fisher information inequality.

Findings

Entropy jump inequality for any pair of log-concave vectors

Fisher information inequality analogous to entropy results

Quantitative Blachmann-Stam inequality established

Abstract

We extend the result of Ball and Nguyen on the jump of entropy under convolution for logconcave random vectors. We show that the result holds for any pair of vectors (not necessarily identically distributed) and that a similar inequality holds for the Fisher information, thus providing a quantitative Blachmann-Stam inequality